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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Assume a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A \sin x.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
-A \sin x-2 A \cos x-3 A \sin x=2 \sin x \to -4 A \sin x-2 A \cos x=2 \sin x \to \textrm{can not be satisfied.}
\end{equation*}
</div>
<p class="continuation">Instead, we assume</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A \sin x+B \cos x.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;-A \sin x-B \cos x-2 (A \cos x-B \sin x)- 3 A \sin x-3 B \cos x= 2 \sin x\\
&amp;\to (-4 A+2 B) \sin x+(-4 B-2 A) \cos x=2 \sin x\\
&amp; \to -4A+2B=2, \quad -4B-2A=0 \\
&amp;\to A=-\frac{2}{5}, \quad B=\frac{1}{5}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=-\frac{2}{5} \sin x+\frac{1}{5} \cos x.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_6.html#p-122" class="internal">in-context</a></span>
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